Average Error: 37.4 → 0.4
Time: 7.9s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{1}{\frac{\left(1 + \cos \varepsilon\right) \cdot \sin x}{{\sin x}^{2} \cdot \left(-{\sin \varepsilon}^{2}\right)}}\right) \]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{1}{\frac{\left(1 + \cos \varepsilon\right) \cdot \sin x}{{\sin x}^{2} \cdot \left(-{\sin \varepsilon}^{2}\right)}}\right)

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma
  (sin eps)
  (cos x)
  (/
   1.0
   (/
    (* (+ 1.0 (cos eps)) (sin x))
    (* (pow (sin x) 2.0) (- (pow (sin eps) 2.0)))))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	return fma(sin(eps), cos(x), (1.0 / (((1.0 + cos(eps)) * sin(x)) / (pow(sin(x), 2.0) * -pow(sin(eps), 2.0)))));
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.4
Target14.8
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 37.4

    \[\sin \left(x + \varepsilon\right) - \sin x \]

  2. Applied sin-sum_binary6422.4

    \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x \]

  3. Applied associate--l+_binary6422.4

    \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \]

  4. Applied associate-+r-_binary6422.4

    \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \]

  5. Simplified22.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \cos \varepsilon \cdot \sin x\right)} - \sin x \]

  6. Applied fma-udef_binary6422.4

    \[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)} - \sin x \]

  7. Applied associate--l+_binary640.4

    \[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \left(\cos \varepsilon \cdot \sin x - \sin x\right)} \]

  8. Applied fma-def_binary640.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \cos \varepsilon \cdot \sin x - \sin x\right)} \]

  9. Applied flip--_binary640.5

    \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\frac{\left(\cos \varepsilon \cdot \sin x\right) \cdot \left(\cos \varepsilon \cdot \sin x\right) - \sin x \cdot \sin x}{\cos \varepsilon \cdot \sin x + \sin x}}\right) \]

  10. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\color{blue}{\left(\sin x \cdot \sin x\right) \cdot \left(-\sin \varepsilon \cdot \sin \varepsilon\right)}}{\cos \varepsilon \cdot \sin x + \sin x}\right) \]

  11. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{\left(\sin x \cdot \sin x\right) \cdot \left(-\sin \varepsilon \cdot \sin \varepsilon\right)}{\color{blue}{\mathsf{fma}\left(\sin x, \cos \varepsilon, \sin x\right)}}\right) \]

  12. Applied clear-num_binary640.4

    \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\sin x, \cos \varepsilon, \sin x\right)}{\left(\sin x \cdot \sin x\right) \cdot \left(-\sin \varepsilon \cdot \sin \varepsilon\right)}}}\right) \]

  13. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{1}{\color{blue}{\frac{\left(1 + \cos \varepsilon\right) \cdot \sin x}{{\sin x}^{2} \cdot \left(-{\sin \varepsilon}^{2}\right)}}}\right) \]

  14. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{1}{\frac{\left(1 + \cos \varepsilon\right) \cdot \sin x}{{\sin x}^{2} \cdot \left(-{\sin \varepsilon}^{2}\right)}}\right) \]

Reproduce

herbie shell --seed 2021215 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))